Convex Duality in Math and Finance
Qiji Zhu, WMU

The title of the talk deliberately coincides with the joint course Dr. Peter Carr and I are co-teaching this semester at Courant. As we approach the end of this semester, it is an appropriate time to report to the department the contents and our thinking in designing this interdisciplinary course. I will explain without assuming prior knowledge what is convex duality and then why it matters in mathematical finance by highlighting three major directions of applications: the duality between utility maximization and martingale measures, dual representation of risk measures, and the duality in the process of delta-hedging contingent claims. Finally, I will report an ongoing joint research project that Dr. Peter Carr and I are conducting on the consistency of generalized convexity of the price of a contingent claim to its terminal payoff and related applications. Surprisingly, our result also implies a similar consistency in the generalized convexity for solutions to the Black-Scholes type partial differential equations. We thank Dr. Goodman and Dr. Lin for stimulating discussions related to this application in PDEs.